Qualitative and quantitative studies of the oxidation of polycrystalline copper (Cu) thin films upon exposure to ambient air conditions for long periods (on the order of several months) are reported in this work. Thin films of Cu, prepared by thermal evaporation, were analyzed by means of X-ray photoelectron spectroscopy (XPS) to gain an understanding on the growth mechanism of the surface oxide layer. Analysis of high-resolution Cu LMM, Cu2p3/2, and O1s spectra was used to follow the time dependence of individual oxide overlayer thicknesses as well as the overall oxide composite thickness. Transmission electron microscopy (TEM) and spectroscopic ellipsometry (SE) were used to confirm the results obtained from XPS measurements. Three main stages of copper oxide growth were observed:  (a) the formation of a Cu2O layer, most likely due to Cu metal ionic transport toward the oxide−oxygen interface, (b) the formation of a Cu(OH)2 metastable overlayer, due to the interactions of Cu ions with hydroxyl groups pr...

Oxidation of Polycrystalline Copper Thin Films at Ambient Conditions

Survey of computed grain boundary properties in face-centered cubic metals: I. Grain boundary energy

An iterative solution method, in the form of a preconditioner for a Krylov subspace method, is presented for the Helmholtz equation. The preconditioner is based on a Helmholtz-type differential operator with a complex term. A multigrid iteration is used for approximately inverting the preconditioner. The choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with Fourier analysis. Multigrid analysis results are verified by numerical experiments. High wavenumber Helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner.

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A Novel Multigrid Based Preconditioner For Heterogeneous Helmholtz Problems

In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods. Simply using a Krylov method is much less effective, especially when the wave number in the Helmholtz operator becomes large, and also algebraic preconditioners such as incomplete LU factorizations do not remedy the situation. Even more powerful preconditioners such as classical domain decomposition and multigrid methods fail to lead to a convergent method, and often behave differently from their usual behavior for positive definite problems. For example increasing the overlap in a classical Schwarz method degrades its performance, as does increasing the number of smoothing steps in multigrid. The purpose of this review paper is to explain why classical iterative methods fail to be effective for Helmholtz problems, and to show different avenues that have been taken to address this difficulty.

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Why it is Difficult to Solve Helmholtz Problems with Classical Iterative Methods

Standard multigrid algorithms have proven ineffective for the solution of discretizations of Helmholtz equations. In this work we modify the standard algorithm by adding GMRES iterations at coarse levels and as an outer iteration. We demonstrate the algorithm's effectiveness through theoretical analysis of a model problem and experimental results. In particular, we show that the combined use of GMRES as a smoother and outer iteration produces an algorithm whose performance depends relatively mildly on wave number and is robust for normalized wave numbers as large as 200. For fixed wave numbers, it displays grid-independent convergence rates and has costs proportional to the number of unknowns.

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A Multigrid Method Enhanced by Krylov Subspace Iteration for Discrete Helmholtz Equations

Multigrid methods are known for their high efficiency in the solution of definite elliptic problems. However, difficulties that appear in highly indefinite problems, such as standing wave equations, cause a total loss of efficiency in the standard multigrid solver. The aim of this paper is to isolate these difficulties, analyze them, suggest how to deal with them, and then test the suggestions with numerical experiments. The modified multigrid methods introduced here exhibit the same high convergence rates as usually obtained for definite elliptic problems, for nearly the same cost. They also yield a very efficient treatment of the radiation boundary conditions.

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Wave-ray multigrid method for standing wave equations.

Measurement of the geometry of triple junctions between grain boundaries in polycrystalline materials generates large sets of dihedral angles from which maps of the grain boundary energy may be extracted. A preliminary analysis has been performed for a sample of magnesia based on a three-parameter description of grain boundaries. An extended form of orientation imaging microscopy (OIM) was used to measure both triple junction geometry via image analysis in the SEM and local grain orientation via electron back scatter diffraction. Serial sectioning with registry of both in-plane images and successive sections characterizes triple junction tangents from which true dihedral angles are calculated. We apply Herring's relation at each triple junction, based on the assumption of local equilibrium at the junction. By limiting grain boundary character to a (three parameter) specification of misorientation for the preliminary analysis, we can neglect the torque terms and apply the sine law to the three boundaries. This provides two independent relations per triple junction between grain boundary energies and dihedral angles. Discretizing the misorientation and employing multiscale statistical analysis on large data sets allows (relative) grain boundary energy as a function of boundary character to be extracted from triple junction geometry. A similar analysis of thermal grooves allows the anisotropy of the surface energy to be measured in MgO.

Extracting Grain Boundary and Surface Energy from Measurement of Triple Junction Geometry

Most technologically useful materials arise as polycrystalline microstructures, composed of a myriad of small crystallites, called grains, separated by their interfaces, called grain boundaries. The orientations and arrangements of the grains and their network of boundaries are implicated in many properties across wide scales, for example, functional properties, like conductivity in microprocessors, and lifetime properties, like fracture toughness in structures. Simulation is becoming an important tool for understanding both materials properties and their processing requirements. Here we offer a consistent variational approach to the mesoscale simulation of these systems subject to the Mullins equation of curvature-driven growth in a two-dimensional setting. The main objective is to provide a calibration for future two-dimensional and three-dimensional efforts. We discuss several novel features of our approach, which we anticipate will render it a flexible, scalable, and robust tool to aid in microstructural prediction. Simulation results offer compelling evidence of the predictability and robustness of statistical properties of large systems, such as grain size distribution and texture, that are of immediate interest in materials science.

A Variational Approach to Modeling and Simulation of Grain Growth

Helmholtz equations with their highly oscillatory solutions play an important role in physics and engineering. These equations present the main computational difficulties typical to acoustic, electromagnetic, and other wave problems. They are often accompanied by radiation boundary conditions and are considered on infinite domains. Solving them numerically using standard procedures, including multigrid, is too expensive. The wave-ray multigrid algorithm efficiently solves the Helmholtz equations and naturally incorporates the radiation boundary conditions. Important accuracy properties of the wave-ray solver are discussed in this paper.

Using various mode analyses, we show that, with the right choice of parameters, this algorithm can obtain an approximation to the differential solution with accuracy that equals the accuracy of the target grid discretization. Moreover, the boundary conditions can be introduced with any desired accuracy. Our theoretical conclusions are confirmed by numerical experiments.

Accuracy Properties of the Wave-Ray Multigrid Algorithm for Helmholtz Equations

A mesoscale, variational simulation of grain growth in two-dimensions has been used to explore the effects of grain boundary properties on the grain boundary character distribution. Anisotropy in the grain boundary energy has a stronger influence on the grain boundary character distribution than anisotropy in the grain boundary mobility. As grain growth proceeds from an initially random distribution, the grain boundary character distribution reaches a steady state that depends on the grain boundary energy. If the energy depends only on the lattice misorientation, then the population and energy are related by the Boltzmann distribution. When the energy depends on both lattice misorientation and boundary orientation, the steady state grain boundary character distribution is more complex and depends on both the energy and changes in the gradient of the energy with respect to orientation.

Irene Livshits

Papers

Wave-ray multigrid method for standing wave equations.

Extracting Grain Boundary and Surface Energy from Measurement of Triple Junction Geometry

A Variational Approach to Modeling and Simulation of Grain Growth

Accuracy Properties of the Wave-Ray Multigrid Algorithm for Helmholtz Equations

Mesoscale Simulation of the Evolution of the Grain Boundary Character Distribution